OPTIMAL MULTIPLE STOPPING AND VALUATION OF SWING OPTIONS
RENÉ CARMONA AND NIZAR TOUZI
A BSTRACT. The connection between optimal stopping of random systems and the theory of the Snell
envelope is well understood, and its application to the pricing of American contingent claims is well
known. Motivated by the pricing of swing contracts (whose recall components can be viewed as contingent claims with multiple exercises of American type) we investigate the mathematical generalization
of these results to the case of possible multiple stopping. We prove existence of the multiple exercise
policies in a fairly general set-up. We then concentrate on the Black-Scholes model for which we give
a constructive solution in the perpetual case, and an approximation procedure in the finite horizon case.
The last two sections of the paper are devoted to numerical results. We illustrate the theoretical results
of the perpetual case, and in the finite horizon case, we introduce numerical approximation algorithms
based on ideas of the Malliavin calculus.
1. I NTRODUCTION
The motivation for the present study comes from the commodity markets, and especially the energy
markets where the lack of standardization and the complexity of many contracts has attracted our
attention. Commodity contracts can be extremely involved and many energy structured products are
truffled with embedded options which are too often neither identified nor priced appropriately.
In this paper we concentrate on the mathematical analysis of options with multiple exercises of
the American type. When embedded in base delivery contracts, these options are sometimes called
swing options. In a vibrant industry sector with a multitude of tailor made contracts negotiated over
the counter, the term swing has been used for many different things, and it is important to specify
what we do have in mind when we use this terminology. The types of contracts containing the swing
options we are considering are described in detail in [2] and [16], and they are slightly different
from the gas sale agreements (GSA) discussed in [10]. These contracts for the covenants are not the
same for the buyer and the seller. The continuous time finance problems considered in this paper, are
addressing the pricing and optimal exercises challenges of the recall options contained in the buyer
side of the swing contracts. To keep things at a reasonable level of complexity, we assume the purpose
of this paper, that the holder of a swing option is given the opportunity to exercise several rights, and
that she has total freedom in the choice of the timing of these exercises. This outstanding feature
of the embedded options is reminiscent of American contingent claims with multiple exercises, and
despite the existence of several numerical pricing algorithms such as [24], [16], [10], or [2] to name
a few, it seems that a rigorous mathematical analysis of the valuation of this specific form of multiple
optionality has not been provided in the existing literature. The purpose of this paper is to fill this gap
by offering a first mathematical analysis of these options with multiple American exercises.
Date: October 29, 2004.
The authors are grateful for many interesting discussions with Bruno Bouchard.
1
2
RENÉ CARMONA AND NIZAR TOUZI
The optimal stopping problem is a ”classic” whose solution had far reaching applications in many
different fields. In its most general form, its mathematical solution is provided by the elegant theory
of the so-called Snell envelope. See for example [23] for a clear presentation in the discrete case,
and [12] for a thorough discussion in the context of optimal control of continuous time stochastic
processes. The reader mostly interested in the application to the pricing of contingent claims with
American exercise is referred to the appendices A and D of [17] for a self contained exposé of the
continuous time theory. Practical applications requiring a generalization of the optimal stopping
problem to possibly multiple stops are plentiful. A version of such a generalization in the discrete
time case goes back to [14] and can be found in the book of Cairoli and Dalang [6].
After a short review of the standard results from optimal stopping which are used in the analysis
of single American exercises, we introduce the inductive hierarchy of Snell envelopes needed in
the multiple exercise case, and we give results on the existence and the characterization of a set of
optimal stopping/exercise times under an assumption which guarantees that all these Snell envelopes
have continuous paths. These results are straightforward generalizations of the classical theory. They
are included for the sake of completeness as a way to introduce the notation. We deliberately excluded
the case where the Snell envelope may present a discontinuity in order to avoid technical difficulties
which are beyond the scope of this paper.
We then concentrate on the geometric Brownian motion framework of the Black-Scholes theory,
and we identify explicitly the solution in the case of perpetual swings generalizing the classical problem of the exercise of a perpetual American put option. In this case the optimal exercise regions form
a strictly decreasing sequence of intervals and the exercise times are given by the hitting times of
the ordered set of their boundaries which we identify. See also [7] for the case of one dimensional
diffusions.
The final section of this paper applies the Monte Carlo approximation method suggested in Lions
and Regnier [19] and further developed by Bouchard and Touzi [5] to our context of multiple stopping.
The above papers make an extensive use of the Malliavin calculus. This is obviously an overkill in
the situation considered in this paper. Indeed, the price process is log-normal in the Black-Scholes
framework, and explicit computations are possible. We review this approach while streamlining its
dependence upon the sophisticated tools of the Malliavin calculus, and by providing a direct and
self-contained account of the various steps.
After this paper was completed, we learned of the existence of [22] where the authors derive
Monte Carlo upper and lower bounds by extending Rogers’ duality strategy to American options with
multiple exercises. Motivated by the pricing of chooser flexible caps, they study what could be viewed
as the buyer side of a swing option.
2. O PTIMAL S TOPPING AND A MERICAN O PTIONS
Let (Ω, F, P) be a complete probability space, and F = {Ft }t≥0 be a filtration satisfying the usual
assumptions, i.e. an increasing right continuous family of sub-σ−algebras of F such that F0 contains
all the P-null sets. We also assume that F0 contains only sets of probability zero or one, and we
denote by S the set of all the F-stopping times. Let X = {Xt }t≥0 be a non-negative F−adapted
process satisfying the following properties:
(1)
the process X is continuous a.s
MULTIPLE AMERICAN EXERCISES
3
and
(2)
E{X̄} < ∞ where X̄ is defined by X̄ = sup Xt .
t≥0
Each random variable Xt has the interpretation of a reward or payoff if we stop the process or if
we use one exercise right at time t. We also fix a time T ∈ (0, ∞] which has the interpretation of
maturity, i.e. the time of expiration of our right to stop the process or exercise. In order to include
both the cases of finite and infinite horizon stopping problems in the same framework, we assume
that:
X· ≡ 0
(3)
on (T, ∞) .
In the infinite horizon case T = +∞, we set
F∞ = σ {∪0≤t<∞ Ft }
and X∞ = lim sup Xt .
t→∞
We denote by ST the collection of all the F−stopping times with values in [0, T ]. Given a stopping
time θ ∈ ST , we denote by Sθ,T the subset of ST consisting of all the stopping times τ ≥ θ a.s.
We now review the notation and the results of the classical treatment of the optimal stopping
problem which we will need in our analysis of the optimal multiple stopping problem associated to
the valuation of swing options. The interested reader can find a more general treatment in [12] and
[17]. In particular, the presentation of [12] avoids the continuity condition (1) used here. In any case,
the classical optimal stopping problem consists in the computation of the supremum
X̂0 =
(4)
sup E{Xτ } .
τ ∈ST
together with the possible characterization of the stopping times at which the supremum is attained.
2.1. Snell Envelope and Optimal Stopping Times. The solution of the optimal stopping problem
(4) is best understood by introducing the family {X̂θ ; θ ∈ ST } of random variables defined by:
(5)
X̂θ = ess supτ ∈Sθ E {Xτ |Fθ }
These random variables satisfies the dynamic programming principle:
o
n
(6)
X̂θ = ess supτ ∈Sθ,T E X̂τ |Fθ ,
for all θ ∈ ST . The F−adapted process X̂ = {X̂t }t≥0 is a super-martingale, which is continuous in
expectation, i.e. E{Xτn } −→ E{Xτ } for all sequences (τn )n≥0 ⊂ S with τn → τ a.s because of (1)
and (2), and hence has a càdlàg modification of the process X̂ which we still denote X̂. It is called
the Snell envelope of the process X. It can be characterized as the smallest super-martingale which
dominates X. Moreover, each random variable X̂θ coincides with the evaluation of the process X̂ at
the stopping time θ, i.e.
X̂θ (ω) = X̂θ(ω) (ω) .
We next define the stopping time
n
o
θ0∗ = inf t ≥ 0; X̂t = Xt .
4
RENÉ CARMONA AND NIZAR TOUZI
By right continuity of the sample paths we have X̂T = XT , so θ0∗ ≤ T a.s., and using the properties
∗
(1)-(2) of the process X, it can be proved that the stopped process X̂ θ0 = {X̂t∧θ0∗ }t≥0 is a martingale.
As a consequence we see that
X̂0 = E{X̂θ0∗ } = E{Xθ0∗ },
which shows that θ0∗ is an optimal stopping time for the problem (4). More generally, for any τ ∈ ST ,
the random variable
n
o
(7)
θτ∗ = inf t ≥ τ ; X̂t = Xt
is less than or equal to T a.s. because X̂T = XT when T < ∞. So it defines a stopping time in Sτ,T
and, by continuity of X, it follows that the stopped process {X̂t∧θτ∗ }t≥τ is a martingale. Therefore:
X̂τ = E{X̂θτ∗ |Fτ } = E{Xθτ∗ |Fτ },
and θτ∗ is an optimal stopping time for the problem (5) in the sense that the essential supremum
appearing in (5) is attained for τ = θτ∗ .
2.2. Doob-Meyer Decomposition and Maximal Lp −Inequality. Since the Snell envelope X̂ of
the process X is a càdlàg supermartingale, it follows from the Doob-Meyer decomposition theorem
together with condition (2) that it admits the representation
X̂t = Mt − At for 0 ≤ t ≤ T ,
where M is a uniformly integrable càdlàg martingale, and A is a non-decreasing F−adapted process
with A0 = 0, and E{AT } < ∞. Since X is continuous, we also have that A is continuous and
Z T
1{X̂t >Xt } dAt = 0 ,
0
See Karatzas and Shreve [17], Theorem D13. A consequence of this result is that a sample path of
the Snell envelope X̂ is continuous if and only if the corresponding sample path of the martingale
part M is continuous. This remark implies that the sample paths of the Snell envelope are almost
surely continuous when the filtration F is generated by a Brownian motion. Indeed, in this case
every martingale has continuous sample paths since it can be represented as a stochastic integral with
respect to the Brownian motion. We shall make use of this result later in the paper.
Whenever the reward process X satisfies the integrability condition:
(8)
E{X̄ p } < ∞ for some p > 1
(which is stronger than condition (2)) we can define the uniform integrable martingale X̄ = {X̄t }0≤t≤T
by X̄t = E{X̄|Ft } for 0 ≤ t ≤ T , and use Doob’s maximal inequality in its Lp form
(
)
(
)
p
p
p
p
E{X̄ p } < ∞ .
E sup X̂t
≤ E sup X̄t
≤
p
−
1
0≤t≤T
0≤t≤T
Hence, the Snell envelope X̂ inherits the Lp -integrability of the maximum from the underlying process X. As a consequence we see that
(9)
the process X̂ is left-continuous in expectation.
MULTIPLE AMERICAN EXERCISES
5
Indeed, if {τn }n≥0 is an increasing sequence of stopping times with τn % τ ∈ S, then, X̂τn −→ X̂τ
a.s. by continuity of X̂. Now, observe that
)
(
E{X̂τpn } ≤ E
sup X̂tp
< ∞,
0≤t≤T
so that the sequence {X̂τn }n≥0 is uniformly integrable, and (9) follows.
2.3. Optimal Multiple Stopping. From now on, we assume that the filtration F satisfies the additional requirements:
The filtration F is left continuous and
(10)
every F−adapted martingale has continuous sample paths.
This condition guarantees that the Snell envelop of a continuous process is continuous. This property
will be needed in the subsequent analysis which involves a composition of Snell envelopes. The case
of discontinuous processes leads to technical problems which are beyond the scope of this paper.
Let X be a non-negative F−adapted process satisfying conditions (1), (2) and (8). Recall that the
finite maturity framework is captured by our convention (3). We let S̄T be the set of all F−stopping
times valued in [0, T ] ∪ {T +}, and set
XT + ≡ 0 .
(11)
Remark 1. In the case F = FX of the completion of the canonical filtration generated by a process
X, FX is left-continuous whenever the process X has left-continuous sample paths.
We fix an integer ` ≥ 1 and a positive constant δ. Here, ` represents the number of recall rights we
can exercise, while δ is the length of the refracting time interval which needs to separate two successive exercises. This assumption on the separation of the exercise times prevents them from bunching
up together on top of the optimal stopping time for the classical case reviewed in the previous section.
But it is important to emphasize that we do not make this assumption for mathematical convenience.
As reported in [16] the existence of such a refracting time period is part of the actual swing contracts
(`)
traded in the energy markets. We shall denote by S̄T the collection of all vectors of stopping times
~τ = (τ1 , . . . , τp ) such that
(12)
τ1 ≤ T a.s. and τi − τi−1 ≥ δ on {τi−1 ≤ T } a.s. for all i = 2, . . . , ` .
Motivated by the valuation of swing options with multiple American exercises, we define the optimal
multiple stopping problem by:
(13)
Z0 = sup E {X~τ } where X~τ =
~
τ ∈S̄ (`)
`
X
Xτi .
i=1
In the above multiple stopping problem, the holder of the option is allowed to exercise her rights to
the reward/payoff given by the process X at ` different times of her choosing. Notice that the holder
of the option can decide not to use all her exercise rights by setting the last stopping times to T +.
Such strategies could be desirable in this context because of the presence of the refracting period δ,
which may lead the investor to sacrifice an exercise right in order to benefit from a potential better
future exercise.
6
RENÉ CARMONA AND NIZAR TOUZI
In order to characterize this multiple optimal stopping problem, we introduce the sequence of Snell
envelopes :
d
(i) for i = 1, . . . , ` ,
Y (0) ≡ 0 and Y (i) = X
where, for each integer i = 1, . . . , `, the i-exercise reward process X (i) is defined by:
n
o
(i)
(i−1)
Xt = Xt + E Yt+δ |Ft , for 0 ≤ t ≤ T − δ ,
and, whenever T < ∞,
(i)
Xt
= X̂t , for T − δ < t ≤ T .
In order to apply the results of the single optimal stopping theory in this context, we need to ensure
that the iterated reward processes X (i) inherit conditions (1), (2) and (8).
Lemma 1. Let us assume that condition (10) holds and that the process X is continuous and satisfies
condition (8). Then, for all i = 1, · · · , `, the process X (i) is continuous in expectation, and satisfies
n
o
p
(i)
E X̄ (i)
< ∞ where X̄ (i) = sup Xt .
0≤t≤T
Proof. Since X (1) = X, the statement of the lemma holds trivially for i = 1. Using the properties
of the Snell envelope recalled in Subsection 2.2, and the properties of the filtration, we see that the
processes X (i) and X̂ (i) inherit the pathwise continuity from X and X̂ (i−1) for every i ≥ 1.
p
We now prove that E X̄ (i) < ∞. We proceed by induction. For i = 1, the result is just a re
p
p
statement of condition (8). So we assume that E X̄ (i−1) < ∞, and we prove that E X̄ (i) < ∞.
The martingale inequality implies that:
(
)
(
) n
o
p
p
p
p
(i−1)
(i−1)
p
E sup Yt
≤ E sup E{X̄
|Ft }
≤
E X̄ (i−1)
< ∞
p−1
0≤t≤T
0≤t≤T
from which we conclude that
n
o1/p
E X̄ (i)p
≤ E{X̄ p }1/p + E
)1/p
(
sup
0≤t≤T
(i−1)p
Yt
< ∞. Next we identify a set of optimal stopping times for the multiple stopping problem. Let us set:
n
o
(`)
(`)
τ1∗ = inf t ≥ 0; Yt = Xt
Observe that τ1∗ ≤ T . Next for 2 ≤ i ≤ `, we define
n
o
(`−i+1)
(`−i+1)
∗
∗ <T } + (T +)1{δ+τ ∗ ≥T } .
(14) τi∗ = inf t ≥ δ + τi−1
; Yt
= Xt
1{δ+τi−1
i−1
(`)
Clearly, τ~∗ = (τ1∗ , . . . , τp∗ ) ∈ ST . We are now ready to prove the main result of this section.
Theorem 1. If we assume that condition (10) holds and that the process X satisfies (1), (2) and (8),
then
(p)
Z0 = Y0 = E{X~τ ∗ } .
MULTIPLE AMERICAN EXERCISES
7
(`)
Proof. Let ~τ = (τ1 , . . . , τp ) be an arbitrary element in ST . For ease of notation, we set τ̄i := τ`−i+1 .
Let us set X (0) ≡ 0. A straightforward inductive argument can be used to prove


`


X
(i)
(15)
E {X~τ } ≤ E Xτ̄i + 1{i<`}
Xτ̄j


j=i+1
for all 0 ≤ i ≤ `. Now, using (15), together with the classical optimal stopping theorem, we see that:
n
o
n
o
n
o
(p)
(p)
(p−1)
∗
∗
E {X~τ } ≤ E Xτ(p)
≤
Y
=
E
X
=
E
X
+
E{Y
|F
}
.
∗
∗
τ1
τ1
0
τ
δ+τ
1
1
(p−1)
{Yt∧τ ∗ ;
2
Next, we notice that the stopped sequence
from the previous inequality that:
1
t > δ + τ1∗ } is a martingale. We then deduce
n
o
(p−1)
≤ E Xτ1∗ + E{Yτ ∗ |Fτ1∗ }
2
o
n
(p−1)
.
= E Xτ1∗ + E Yτ ∗
(p)
E {X~τ } ≤ Y0
2
By repeatedly using the above argument, we see that
(p)
E {X~τ } ≤ Y0 ≤ E Xτ1∗ + . . . + Xτ2∗ ,
proving the optimality of the vector of stopping times (τ1∗ , . . . , τp∗ ) for the problem Z0 , together with
(p)
the equality Z0 = Y0 . We next introduce the stopping times
(i)
∗
ζi∗ = inf{ t ≥ δ + τi−1
; Yt
(i)
∗ <T } + (T +)1{τ ∗ ≥T } .
= Xt } 1{τi−1
i−1
∗ < T }. In fact, the stopping
∗
on {τi−1
Using the definition of θτ∗ given in (7), we see that ζi∗ = θδ+τ
∗
i−1
time ζi∗ is the first optimal stopping time for the single exercise/stopping problem (p = 1) starting
∗ , where τ ∗ is an optimal stopping time for the exercise of the (i + 1)−th right.
from time δ + τi+1
i+1
Our next result confirms the intuitive belief that one should exercise earlier if one has more than one
right.
Proposition 1. For any integer i = 1, . . . , p, we have :
(i)
(1)
Yt
≤ Yt
(i)
(1)
(i−1)
t ≥ 0, and τi∗ ≤ ζi∗ .
+ E{Yt+δ |Ft } ,
(i−1)
Proof. Let us set Vt = Yt + E{Yt+δ |Ft }. From the supermartingale property of the processes
Y (1) and Y (i−1) , together with the inequality Y (1) ≥ X, we see that V (i) is a supermartingale which
dominates X (i) . We then conclude that Y (i) ≤ V (i) , whence
(i)
Xt
(i)
(i)
(i−1)
(i)
= Xt + E{Yt+δ |Ft } ≤ Yt
(1)
≤ Yt
(i−1)
+ E{Yt+δ |Ft }
We get Xζ ∗ = Yζ ∗ as an easy consequence of this inequality, and therefore we can conclude that τi∗
i
i
≤ ζi∗ by definition of τi∗ . 8
RENÉ CARMONA AND NIZAR TOUZI
We conclude this section by examining the case where the reward process X is a submartingale. It
is well-known that this implies that the maturity T is an optimal stopping time for the (single) optimal
stopping problem. This applies for instance to call options, and provides the equivalence between
European and American call options, when the underlying asset does not deliver any dividends.
Proposition 2. Assume that X is a submartingale, and let T < ∞ be a finite maturity date. Then
X̂t` =
`
X
E {XT −iδ } 1t≤T −iδ
i=1
(i)
Proof. It is sufficient to notice that for i = 1, . . . , ` and t ≤ T − iδ, we have Xt
(i−1)
E{X̂t+δ |Ft } inherits the submartingale property of X. = Xt +
3. P ERPETUAL S WING O PTIONS IN THE B LACK -S CHOLES F RAMEWORK
In the previous section, we proved the existence of an optimal vector of stopping times, and we
outlined an algorithm to construct such a vector by considering a cascade of Snell envelopes. There is
no hope to get a more explicit characterization in such a generality. In this section, we concentrate on
the problem of the optimal exercise of perpetual (i.e. T = +∞) swing options in the Black-Scholes
framework, and we identify the optimal exercise times as hitting times of a set of thresholds for which
we provide a constructive algorithm.
3.1. The Black-Scholes Set-up. Let W be an R-valued Brownian motion on the probability space
(Ω, F, P), and denote by F = {Ft }t≥0 the associated completed filtration. Notice that F satisfies
condition (10). Let us fix a reward/payoff function φ : R+ −→ R+ and let us consider a reward/payoff
process of the form φ(Xt ) where the process X = {Xt }t is now defined by:
σ2
t + σWt ,
(16)
t≥0
Xt = X0 exp r −
2
where r and σ are two positive parameters standing for the short interest rate and the volatility. Notice
that we changed the notation and that the process X is no longer the reward/payoff process. It is now
the price process from which the reward/payoff φ(X) is computed. We shall also use the notation
Xt0,X0 for Xt whenever we need to emphasize the dependence of the process X upon its initial
condition. We now define the value function of the perpetual swing option problem with ` exercise
rights and refraction time δ > 0 by:
( `
)
X
(17)
v (`) (X0 ) =
sup
E
e−rτi φ(Xτi ) .
(τ1 ,...,τ` )∈S (`)
i=1
where the set S (`) is defined by:
n
o
S (`) =
~τ = (τ1 , . . . , τ` ) ∈ S ` ; τi − τi−1 ≥ δ for all i = 2, . . . , ` ,
and S is the subset of S∞ of all finite stopping times with values in R+ . In the following, we restrict
ourselves to the reward function
(18)
φ(x) = (K − x)+ for some given parameter K > 0 ,
MULTIPLE AMERICAN EXERCISES
9
so that our analysis of the optimal exercise of the swing option extends to the multiple exercise case,
classical results proven in the case of an American put option with only one exercise right.
3.2. Perpetual American Puts. When ` = 1 the refracting time parameter δ is irrelevant. The following explicit solution of the perpetual American put option is well-known. We state it to introduce
the notation we need in the sequel.
Theorem 2. The perpetual American put option value function is given by
∗ γ x1
(1)
∗
v (x) = (K − x ∧ x1 ) 1 ∧
,
x
where
γ =
2r
Kγ
and x∗1 =
.
2
σ
1+γ
The above result states in particular that
v (1) (x) = K − x = φ(x) if and only if x ≤ x∗1 .
The perpetual American put option should be exercised whenever the price process X is below the
level x∗1 . In view of this, the interval [0, x∗1 ] is called the exercise region, and its complement (x∗1 , ∞)
is called the the continuation region.
In the subsequent paragraphs, we provide an extension of Theorem 2 to the case of perpetual options with multiple exercises. We first prove the existence of a non-empty and connected exercise
region [0, x∗` ] for the optimal multiple stopping problem with value function v (`) . This region contains the exercise region [0, x∗1 ] of the perpetual American put option. We then provide a recursive
characterization of the boundary x∗` as the unique zero of some functional involving the value function
v (`−1) . As a by-product, this characterization produces a quasi-constructive solution of the problem
of interest, and allows us to prove that the sequence (x∗` )` is increasing.
3.3. Single Stopping Time Formulation and ODE Characterization. We first apply the general
characterization result of Theorem 1 to reduce the multiple stopping problem to a cascade of p optimal
single stopping problems. So we define inductively the value functions v (k) and the reward/payoff
functions φ(k) by setting:
n
o
(19)
v (k) (x) = sup E e−rτ φ(k) Xτ0,x
,
τ ∈S
where
n
o
φ(k) (x) = φ(x) + e−rδ E v (k−1) (Xδ0,x ) ,
i.e. the process e−rt v (k) (Xt ) is the continuous-time Snell envelope of the process e−rt φ(k) (Xt ) .
One can then write the dynamic programming principle for the single stopping time problem value
function (19), and derive the so-called Hamilton-Jacobi-Bellman equation for v (k) . Using classical
analysis arguments, one can prove that this function is the unique continuous viscosity solution of the
variational inequality
n
o
min −Lv (k) , v (k) − φ(k) ≥ 0 ,
10
RENÉ CARMONA AND NIZAR TOUZI
where L is the linear second order differential operator
1 2 2 ∂2χ
∂χ
σ x
(x) + rx (x) − rχ(x) .
2
∂x2
∂x
We shall not provide a proof of this claim because we do not use the above PDE characterization of
v (k) in what follows.
Since φ is bounded and X is continuous, the reward process {e−rt φ(Xt )}t≥0 satisfies the conditions (1) and (8), and we can apply the results of the previous section. For later use, we provide the
following well-known result (which obviously holds for more general diffusions), which gives among
other things, the fact that the reward functions φ(k) are continuous.
Lχ(x) =
Lemma 2. Let f : [0, ∞) −→ R be such that E{|f (Xt )|2 } < ∞, and let us consider the function
n
o
χ(x) = E f (Xt0,x ) , x ≥ 0.
Then χ is continuous on [0, ∞) and continuously differentiable on (0, ∞).
Proof. We sketch the proof for the sake of completeness. Observe that:
Z 1
2
2
2
√
(20)
χ(x) =
f ze(r−σ /2)t
e−(ln z−ln x) /2σ t dz .
zσ 2πt
The dominated convergence theorem allows differentiation inside the expectation and the desired
result follows. 3.4. First Properties of Perpetual Swing Options. We first prove that the exercise region for the
optimal stopping problem (19) is not empty.
Lemma 3. Let k ≥ 1 be a given integer and let φ(x) = (K − x)+ . Then
v (k) (x) = φ(k) (x) whenever 0 ≤ x ≤ x∗1 .
Proof. We only need to prove that v (k) ≤ φ(k) on [0, x∗1 ] as the reverse inequality is trivial. By
definition of φ(k) , we have
n
o
v (k) (x) = sup E e−rτ φ(Xτ ) + e−r(τ +δ) v (k−1) (Xτ +δ )
τ ∈S
n
o
≤ v (1) (x) + sup E e−r(τ +δ) v (k−1) (Xτ +δ ) .
τ ∈S
(21)
Notice that E e−r(τ +δ) v (k−1) (Xτ +δ ) ≤ E e−rδ v (k−1) (Xδ ) by the supermartingale property of
the process {e−rt v (k−1) (Xt )}t≥0 . Since v (1) = φ(1) = φ on [0, x∗1 ], this provides :
n
o
v (k) (x) ≤ φ(x) + E e−rδ v (k−1) (Xδ ) = φ(k) (x) for 0 ≤ x ≤ x∗1 . Our next result shows that the exercise region corresponding to the stopping problem (19) is connected, and defines the exercise boundaries x∗k .
MULTIPLE AMERICAN EXERCISES
11
Proposition 3. Let k ≥ 1 be a given integer and as above, let us restrict ourselves to the payoff
φ(x) = (K − x)+ of the American put option. Then, there exists x∗k ∈ [x∗1 , K] such that
v (k) (x) = φ(k) (x) if and only if 0 ≤ x ≤ x∗k .
Proof. Let us assume to the contrary that the exercise region for the problem v (k) is not connected,
and we will establish a contradiction. Let 0 ≤ x1 < x2 be such that
(22)
v (k) (xi ) = φ(k) (xi ) while v (k) > φ(k) on (x1 , x2 ) .
Since φ(x) = 0 for x ≥ K, it follows that x2 ≤ K, and therefore
φ(x) = K − x for x ∈ [x1 , x2 ] .
(23)
Setting
o
n
x1 + x2
, Sxi = inf t ≥ 0; Xt0,x̂ = xi
2
it follows from the classical optimal stopping theory that the stopping time Ŝ = Sx1 ∧ Sx2 is optimal
for the problem v (k) (x̂). See Subsection 2.1. By (23), we then see that :
o
o
n
n
+ E e−r(Ŝ+δ) v (k−1) X 0,x̂
v (k) (x̂) = E e−rŜ φ X 0,x̂
Ŝ+δ
Ŝ
o
n
0,x̂
0,x̂
= φ E{e−rŜ X } + E e−r(Ŝ+δ) v (k−1) X
Ŝ+δ
Ŝ
n
o
0,x̂
= φ(x̂) + E e−r(Ŝ+δ) v (k−1) (X
)
Ŝ+δ
o
n
(24)
= φ(k) (x̂) ,
≤ φ(x̂) + E e−rδ v (k−1) Xδ0,x̂
o
n
where the last inequality follows from the super-martingale property of e−rt v (k−1) (Xt0,x̂ )
. This
x̂ =
t≥0
proves that v (k) (x̂) = φ(k) (x̂) contradicting (22). In view of this result, we conclude that the stopping time:
θk∗ = inf{t ≥ 0; Xt ≤ x∗k }
defines an optimal stopping rule for the problem (19), and that the value function of the problem is
given by:
n
o
n
o
∗
∗
(25)
v (k) (x) = E e−rθk φ(k) (Xθk∗ ) = φ(k) (x ∧ x∗k ) E e−rθk
by the continuity of the process X.
3.5. Characterization of the Exercise Boundaries of Perpetual Put Swing Options. So far, we
have proved that the multiple stopping problem given by the value function v (`) could be reduced to
a cascade of single optimal stopping problems with value functions v (k) , k ≤ ` in such a way that the
optimal stopping/exercise rules were given by the exit times of the intervals [x∗k , ∞). It is therefore
natural to introduce for each (reward) function ψ the function w[ψ] defined by:
n
o
n
o
x
0,x
x
(26)
)
where
S
=
inf
t
≥
0;
X
≤
b
,
w[ψ](x, b) := E e−rSb ψ(XS0,x
x
b
t
b
12
RENÉ CARMONA AND NIZAR TOUZI
which we will use for ψ = φ(k) , and we make the following important observation:
(k)
φ (x) for x ≤ x∗k
(k)
v (x) =
(27)
w[φ(k) ](x, x∗k ) = maxb≤x w[φ(k) ](x, b) for x ≥ x∗k .
In order to compute explicitly the function w[φ(k) ], we use the following well-known property of the
Brownian motion. For each b ≥ 0, let us define by:
Tb = inf{t ≥ 0; µt + Wt = b}
(28)
the first hitting time of the barrier b by the Brownian motion with drift µ. It is well-known that the
Laplace transform of Tb is given by
√
n
o
2
(29)
E e−λTb
= eµb−|b| µ +2λ .
Let
Then Sbx = Tβ(b,x)
R, we have
σ
1
b
µ = (γ − 1) and β(b, x) =
ln
.
2
σ
x
1{x≥b} and it follows from (29) that for any bounded measurable function ψ: R+ −→
n
o
x
w[ψ](x, b) = E e−rSb ψ XS0,x
x
b
= ψ(x) 1{x≤b} + ψ(b)E e−rTβ(b,x) 1{x>b}
γ b
.
= ψ(x ∧ b) 1 ∧
x
(30)
Plugging this equality in (27) and recalling that x∗k ≤ K, we see that

o
n
 K − x + e−rδ E v (k−1) (X 0,x ) for x ≤ x∗
k
δ
n
o
(31) v (k) (x) =
 x−γ maxb≤K bγ K − b + e−rδ E v (k−1) (X 0,b )
for x ≥ x∗k .
δ
In particular, if we use Lemma 2, this shows that the function v (k) is differentiable on (0, x∗k ) ∪
(x∗k , ∞). Since 0 < x∗k < K, the boundary x∗k is the solution of the equation given by the first order
condition for the above maximization problem. For the statement of this result, it is convenient to
introduce the function
d (k)
(k)
−1
(k)
(32)
u (x) = (1 + γ)
γv (x) + x v (x) , x > 0 .
dx
Observe that v (k) can be recovered from u(k) by
v
(33)
(k)
−γ
Z
(x) = (1 + γ)x
x
y γ−1 u(k) (y)dy
0
and
v (k) (0)
= φ(0)
Pk−1
i=0
e−irδ ,
as 0 is an absorbing boundary for the process X.
Lemma 4. For each integer k ≥ 1, the function u(k) is non-increasing and continuous. Moreover the
sequence {u(k) }k is characterized by the induction formula:
n
o
(34)
u(k) (x) = 1{x≤x∗k } x∗1 − x + e−rδ E u(k−1) (Xδ0,x )
and u(0) = 0 .
MULTIPLE AMERICAN EXERCISES
13
Furthermore, the boundary x∗k is uniquely defined by the equation:
(35)
n
o
0,x∗
x∗1 − x∗k + e−rδ E u(k−1) (Xδ k )
= 0.
Proof. The first order condition (35) is obtained by direct differentiation. In order to prove (34), we
(k)
use the expression of v (k) given in (31). First, for x ≥ x∗k , it is clear
n that u (x) =
o 0. Next, since
Xδ0,x = xXδ0,1 , we see immediately that u(k) (x) = x∗1 − x + e−rδ E u(k−1) (Xδ0,x ) for x < x∗k .
The function u(k) is clearly continuous away from the point x∗k . Recall Lemma 2. Using the first
order condition (35), we see that u(k) (x∗k −) = u(k) (x∗k +) = 0.
We now prove by induction that u(k) is non-increasing. This property holds trivially for u(0) ≡ 0.
Assume that u(k−1) is non-decreasing for some k ≥ 2, then since Xδ0,x is increasing in x, it follows
from (34) that u(k) is non-increasing. Finally, we observe that the uniqueness of the exercise boundary
x∗k follows from the fact that the function u(k) is non-increasing. Remark 2. Formula (34) is particularly well suited for numerical computations. Indeed it is plain
to evaluate the expectation by a simple Monte Carlo method. We used this remark to produce the
numerical results reported Section 5. See Figures 1, 2 and 3 for plots of the functions u(k) .
Recall that Proposition 3 says that x∗k ≥ x∗1 . It is natural to expect that these exercise boundaries
form a monotone sequence. We establish this highly expected increasing property of the sequence
(x∗k )k≥1 in the present context.
Lemma 5. (i) The sequence (u(k) )k≥0 is increasing, i.e. u(k) ≥ u(k−1) and u(k) 6= u(k−1) .
(ii) The sequence (x∗k )k≥1 is strictly increasing.
o
n
Proof. Define the functions `(x) = x − x∗1 and g (k) (x) = e−rδ E u(k) (Xδ0,x ) . We first prove that
(ii) is a direct consequence of (i). Indeed, it follows from (i) that g (k) (x) > g (k−1) (x) for all x > 0.
Since x∗k is the unique intersection point of the graphs of the functions ` and g (k−1) , we conclude that
x∗k > x∗k−1 .
We prove (i) by an induction argument. We first remark that u(1) (x) = (x∗1 − x)+ for all x ≥ 0.
Since u(0) ≡ 0, this shows that u(1) ≥ u(0) and u(1) 6= u(0) . We next assume that u(k−1) ≥ u(k−2)
and that u(k−1) 6= u(k−2) . Observe that this implies that x∗k ≥ x∗k−1 by the first part of this proof.
Moreover
• For x > x∗k , we have u(k) (x) = u(k−1) (x) = 0.
• For x < x∗k−1 , we have
n
o
[u(k) − u(k−1) ](x) = e−rδ E [u(k−1) − u(k−2) ](Xδ0,x ) > 0 .
• For x ∈ [x∗k−1 , x∗k ], we have u(k−1) (x) = 0. Furthermore, since u(k) is non-increasing and
u(k) (x∗k ) = 0, it follows that u(k) (x) ≥ 0. 14
RENÉ CARMONA AND NIZAR TOUZI
4. P ERPETUAL P UT S WING O PTIONS WITH I NFINITELY M ANY E XERCISE R IGHTS IN THE
B LACK -S CHOLES M ODEL
In this section we study the asymptotic regime obtained when the number of exercise rights increases without bound. In order to do so, we analyze the value function:



X
(36)
e−rτn φ (Xτn ) ,
v (∞) (X0 ) =
sup
E


(τn )n≥1 ∈S (∞)
n≥1
where as before φ is the payoff φ(x) = (K − x)+ of an American put option and where the set S (∞)
of sequences of stopping times depends upon the refraction parameter δ > 0 in the following way:
n
o
S (∞) = (τn )n≥1 ∈ S N ; τn+1 − τn ≥ δ for all n ≥ 1 .
Observe that, since φ ≤ K, we have for all (τn )n≥1 ∈ S (∞) that:
X
X
X
e−rτn φ (Xτn ) ≤ K
e−rτn ≤ K
e−rnδ .
n≥1
n≥1
n≥0
Therefore:
−1
.
v (∞) (X0 ) ≤ K 1 − e−rδ
(37)
We recast the current problem in the framework of an optimal single stopping problem.
Proposition 4. The value function v (∞) satisfies:
n
o
n
o
(38) v (∞) (X0 ) = sup E φ(∞) (Xτ )
where φ(∞) (x) = φ(x) + e−rδ E v (∞) (Xδ0,x ) .
τ ∈S
Proof. 1. Let τ ∈ S and (τ̄i )i≥1 ∈ S (∞) , and let us define a new sequence {τi }i≥1 of stopping times
by τ1 = τ , τi = τ + δ + τ̄i+1 . Then (τi )i≥1 ∈ S (∞) . Therefore


X

v (∞) (X0 ) ≥ E
e−rτi φ(Xτi )


i≥1




X

= E e−rτ φ(Xτ ) + e−r(τ +δ) E
e−rτ̄i φ(Xτ +δ+τ̄i )|Fτ +δ
.



i≥1
Since the sequence {τ̄i }i≥1 ∈ S (∞) and the stopping time τ ∈ S were arbitrary, this gives the
inequality:
n
o
v (∞) (X0 ) ≥ sup E e−rτ φ(Xτ ) + e−r(τ +δ) v (∞) (Xτ +δ ) .
τ ∈S
MULTIPLE AMERICAN EXERCISES
15
2. In order to establish the reverse inequality, we pick ε > 0 and an ε-optimal stopping rule {τ̂i }i≥1 ∈
S (∞) , i.e.


X

e−rτ̂i φ(Xτ̂i )
v (∞) (X0 ) − ε ≤ E


i≥1




X
(39)
e−rτ̄i φ (Xτ̂1 +δ+τ̄i ) ,
= E e−rτ̂1 φ (Xτ̂1 ) + e−r(τ̂1 +δ)


i≥1
where we used the stopping times τ̄i defined by:
τ̄i = τ̂i+1 − τ̂1 − δ for i ≥ 1 .
We now observe
that τ̄1 ≥ 0 and τ̄i+1 − τ̄i ≥ δ a.s. Furthermore, considering the shifted filtration
n
o
τ̂1 +δ
τ̂
+δ
1
F
= Ft
, we see that τ̄i is an Fτ̂1 +δ −stopping time, and therefore
t≥0


X

E
e−rτ̄i φ (Xτ̂1 +δ+τ̄i ) Fτ̂1 +δ
≤ v (∞) (Xτ̂1 +δ ) ,


i≥1
by definition of the value function v (∞) . Plugging this inequality in (39), we deduce that
n
o
v (∞) (X0 ) − ε ≤ E e−rτ̂1 φ (Xτ̂i ) + e−r(τ̂1 +δ) v (∞) (Xτ̂1 +δ )
n
o
≤ sup E e−rτ φ (Xτ ) + e−r(τ +δ) v (∞) (Xτ +δ ) . τ ∈S
Next, we proceed exactly as in the proofs of Lemma 3 and Proposition 3 to obtain the following
result.
Proposition 5. There exists x∗∞ ∈ [x∗1 , K] such that
v (∞) (x) = φ(∞) (x) if and only if 0 ≤ x ≤ x∗∞ .
Consequently, the stopping time
(40)
∗
θ∞
= inf {t ≥ 0 : Xt ≤ x∗∞ }
defines an optimal stopping rule for the problem (36). In order to characterize further the boundary
x∗∞ , we proceed as in the previous section by observing that
(∞)
φ (x) for x ≤ x∗∞
(∞)
v (x) =
w[φ(∞) ](x, x∗∞ ) = maxb≤x w[φ(∞) ](x, b) for x ≥ x∗∞ ,
which provides by the same computations:

n
o
 K − x + e−rδ E v (∞) (X 0,x ) for x ≤ x∗∞
δ
n
o
(41) v (∞) (x) =
 x−γ maxb≤K bγ K − b + e−rδ E v (∞) (X 0,b )
for x ≥ x∗∞ .
δ
16
RENÉ CARMONA AND NIZAR TOUZI
n
o
Recall from (37) that v (∞) is bounded. Then the function x 7−→ E v (∞) (Xδ0,x ) is differentiable by
Lemma 2. Therefore, we deduce from (41) that v (∞) is differentiable on [0, x∗∞ ) ∪ (x∗∞ , ∞). Since
0 < x∗∞ < K, the boundary x∗∞ solves the first order condition of the above maximization problem.
As in the previous section, it is convenient to introduce the function:
d (∞)
(∞)
(∞)
−1
γv (x) + x v (x) , x > 0 ,
u (x) = (1 + γ)
dx
and we observe that v (∞) can be recovered from u(∞) by
Z x
(∞)
(∞)
−γ
y γ−1 u(k) (y)dy
v (x) = v (0) + (1 + γ)x
0
Z x
−1
−γ
−rδ
(42)
y γ−1 u(∞) (y)dy
+ (1 + γ)x
= K 1−e
0
We now state the first order conditions for the optimality of x∗∞ .
Lemma 6. The function u(∞) is continuous and satisfies
o
n
(43)
,
u(∞) (x) = 1{x≤x∗∞ } x∗1 − x + e−rδ E u(∞) (Xδ0,x )
and the boundary x∗∞ solves
(44)
n
o
0,x∗
x∗1 − x∗∞ + e−rδ E u(∞) (Xδ ∞ )
= 0.
Proof. Except the continuity of u(∞) , all claims follow by the same arguments as
n in the proofo of
(∞)
Lemma 4. Now, since v
is bounded by (37), observe that the function x 7−→ E v (∞) (Xδ0,x ) is
continuously differentiable. By (41), this shows that v (∞) is continuously differentiable on (0, x∗∞ ),
implying the continuity of u(∞) on this set. Since u(∞) = 0 on (x∗∞ , ∞), it only remains to check the
continuity of u(∞) at the point x∗∞ . This is a direct consequence of the first order condition (44). We are now in a position to characterize the boundary x∗∞ in terms of the integer-valued random
variable Nδ (b) defined as the first crossing time of the geometric Brownian motion samples separated
by the refracting period, i.e. the integer:
n
o
0,1
Nδ (b) := min n ≥ 1; Xnδ
>b .
(45)
Proposition 6. The function u(∞) is given by:


∗ /x)−1
∞
Nδ (xX

0,x
u(∞) (x) = E
e−nrδ x∗1 − Xnδ
(46)
,


n=0
and the the optimal exercise boundary for the problem (36) is :
P
−nrδ }
E{
n<Nδ (1) e
∗
∗
x∞ = x1
(47)
P
0,1 .
E{ n<Nδ (1) e−nrδ Xnδ
}
MULTIPLE AMERICAN EXERCISES
17
Q
Proof. Iterating (43), and observing that ji=0 1{X 0,x ≤x∗ } = 1{j<Nδ (x∗∞ /x)} , we see that
1
iδ


n
X

0,x
u(∞) (x) = E
e−jrδ x∗1 − Xjδ
1{j<Nδ (x∗∞ /x)}


j=0
n
o
0,x
+ e−(n+1)rδ E u(∞) (X(n+1)δ
)1{n<Nδ (x∗∞ /x)} ,
by definition of Nδ (b). Now, recall that u(∞) = 0 outside the compact interval [0, x∗∞ ] and that it is
continuous by Lemma 6. Then u(∞) is bounded and
n
o
0,x
lim e−(n+1)rδ E u(∞) X(n+1)δ
1{n<Nδ (x∗∞ /x)} = 0 .
n→∞
By the dominated convergence theorem, this implies that :


n
X

0,x
1{j<Nδ (x∗∞ /x)}
u(∞) (x) = lim E
e−jrδ x∗1 − Xjδ
n→∞


j=0


∞
X

0,x
= E
e−jrδ x∗1 − Xjδ
1{j<Nδ (x∗∞ /x)} ,


j=0
completing the proof of (46). We now obtain (47) by writing that u(∞) (x∗∞ ) = 0:


(1)−1
NδX

0,1
0 = E
e−nrδ x∗1 − x∗∞ Xnδ


n=0




(1)−1
(1)−1
NδX

NδX

0,1
= x∗1 E
e−nrδ − x∗∞ E
(48)
e−nrδ Xnδ
.




n=0
n=0
We conclude this section by proving the convergence of the solution of the perpetual swing option
problem with finitely many rights to the corresponding solution when the number of exercise rights
is infinite.
Proposition 7. We have the following convergence results:
(49)
x∗k −→ x∗∞ and
u(k) , v (k) −→ u(∞) , v (∞) uniformly on [0, ∞) .
Proof. 1. We first prove the convergence of {x∗k }k≥1 and the uniform convergence of {u(k) }k≥1 .
By Lemma 5, the sequences {x∗k }k≥1 and {u(k) }k≥1 are increasing. Moreover x∗k ≤ K, and u(k) ≤
K[1 − e−rδ ]−1 as it can be immediately checked from (34). Then there exist x̄ ≤ K and ū :
[0, ∞) −→ R such that
x∗k −→ x̄ and u(k) (x) −→ ū(x) for all x ≥ 0 .
By dominated convergence, it follows from (34) and (35) that
n oo
n
(50)
ū(x) = 1x≤x̄ x∗1 − x + e−rδ E ū Xδ0,x
18
RENÉ CARMONA AND NIZAR TOUZI
and
(51)
n o
x∗1 − x̄ + e−rδ E ū Xδ0,x̄
= 0.
Since ū is bounded, this proves that ū is continuous. Since, for each k ≥ 1, u(k) = 0 outside the
compact interval [0, K], it then follows from Dini’s theorem that {u(k) }k≥1 converges to ū uniformly
on [0, ∞).
2. We now show that (x̄, ū) = (x∗∞ , u(∞) ). To see this, notice that (50) and (51) show that (x̄, ū)
satisfy the conditions which have been established for (x∗∞ , u(∞) ) in Lemma 6. Observing that the
characterization of (x∗∞ , u(∞) ) in Proposition 6 is obtained by means of these equations, we conclude
that (x̄, ū) = (x∗∞ , u(∞) ).
3. It remains to prove the uniform convergence of sequence v (k) k≥1 towards v (∞) . To see this, we
directly compute by (33) and (42) that :
Z x
(k)
(k)
γ−1 (k)
(∞)
(∞)
(∞)
−γ
y
u
(y)
−
u
(y)
v
(x)
−
v
(x)
≤
v
(0)
−
v
(0)
+
(1
+
γ)x
dy
0
Z x
(k)
(∞)
(k)
(∞)
−γ
≤ v (0) − v (0) + (1 + γ)ku − u k∞ x
y γ−1 dy
0
= v (k) (0) − v (∞) (0) + γ −1 (1 + γ)ku(k) − u(∞) k∞ ,
where we used the notation k · k∞ for the supremum norm of a function. The desired result then
follows from the uniform convergence of {u(k) }k≥1 towards u(∞) . 5. N UMERICAL R ESULTS FOR THE I NFINITE M ATURITY P ROBLEM
This section contains a small sample of numerical results chosen to illustrate the theoretical results
proven in this paper. The computations reported in this section use a strike price K = 1, a refraction
period δ = .01, a volatility σ = .35, and a short interest rate r = .04. These give the values γ = .653
and x∗1 = .395.
We computed approximations of the function uk over a grid of 250 points x regularly spaced
between 0 and 1. We used the values 1, 2, · · · , 250 for k. The expectation appearing in the recursive
formula (34) proven in Lemma 4 and giving the values of uk in terms of uk−1 was computed from
20, 000 Monte Carlo random samples of the random variable Xδ starting from the values X0 = x of
the grid.
Figure 1 shows the graphs of the first 250 functions uk computed in this way. One clearly sees the
fact that they are increasing. This growth in k near the origin could appear in contradiction with the
fact that the uk ’s were uniformly bounded (i.e. the supremum norm of uk is bounded in k) which we
proved in the text. So we plotted these supremum norms as functions of k.
Figure 2 is very much consistent with the uniform boundedness of the functions uk .
This is confirmed by the surface plot given in Figure 3 which gives a different perspective on the
same data. Finally, Figure 4 plots the values of x∗k as a function of k. After an early sharp increase,
these thresholds level off rapidly toward their limiting value x∗∞ .
MULTIPLE AMERICAN EXERCISES
19
F IGURE 1. Time series plots of the graphs of the functions uk for k = 10j with
j = 1, 2, · · · , 25. Note that the numerical value of x∗∞ would be .725.
F IGURE 2. Time series plot of the values of the sup-norm of the function uk for k = 1, 2, · · · , 250.
√
Figure 5 gives a surface plot of x∗∞ as a function
of
the
the
two
free
parameters
rδ
and
σ
δ/2.
√
√
Notice that x∗∞ is found to be decreasing in σ δ/2 and increasing in r δ, both properties being very
natural.
20
RENÉ CARMONA AND NIZAR TOUZI
F IGURE 3. Surface plot of the values uk (x) for x ∈ [0, 1] and for k = 1, 2, · · · , 250..
6. N UMERICAL R ESULTS FOR THE F INITE H ORIZON P ROBLEM
In this section, we return to the finite horizon multiple stopping problem, and we set
T =1
without any loss of generality. In other words we consider swing options with maturity one year. Our
objective is to present and implement a Monte Carlo numerical procedure for the computation of the
value function of the multiple stopping problem
(52)
v (`) (0, X0 ) :=
sup
`
X
E e−rτi φ (Xτi ) ,
~
τ ∈S̄ (`) i=1
and the associated exercise region. As before, X is the Black-Scholes price process defined in (16), r
is a constant instantaneous interest rate, and φ(x) = (K − x)+ is the European put pay-off function
with strike K > 0.
6.1. Discrete Time Approximation. In order to estimate the value function v (`) , we first need to
define a convenient discrete-time approximation. For each integer n ≥ 1, we introduce the partition
Tn = {tj := j/n}0≤j≤n of the time interval T := [0, 1], and we use the notation Sn for the subset
MULTIPLE AMERICAN EXERCISES
21
F IGURE 4. Plot of the values of x∗k computed for k = 1, 2, · · · , 250.
of S1 defined by
Sn := {τ ∈ S1 ; τ ∈ Tn a.s.} .
We use the same notation as in Subsection 2.3. In particular, Y (i) stands for the Snell envelope of
the reward process φ(i) (t, Xt ), where the reward function φ(i) is defined inductively together with the
successive value functions v (i) by:
n
o
(53)
φ(i) (t, x) := φ(x) + e−rδ E v (i−1) t + δ, Xδ0,x
for t ≤ 1 − δ ,
and
φ(i) (t, x) := φ(x)
(54)
for 1 − δ < t ≤ 1 ,
while the v (i) ’s are given by:
(55)
v (i) (t, x) =
n
o
sup E e−rτ φ(i) τ, Xτt,x .
τ ∈St,T
Recall the convention v (0) ≡ 0. For each integer n ≥ 1, we propose a natural discrete time approximation for the value function of the problem. It is given by
h
i
t,x
(56)
vn(i) (t, x) :=
sup
E e−rτ φ(i)
τ,
X
,
n
τ
τ ∈Sn ∩St,T
(0)
starting as before with vn ≡ 0, and where:
n
o
0,x
−rδ
(i−1)
(57)
φ(i)
(t,
x)
=
φ(x)
+
e
E
v
t
+
δ,
X
n
n
δ
for t ≤ 1 − δ ,
22
RENÉ CARMONA AND NIZAR TOUZI
F IGURE √
5. Surface plot of the values of x∗∞ as function of the two free parameters
rδ and σ δ/2.
and
φ(i)
n (t, x) = φ(x)
(58)
for 1 − δ < t ≤ 1 .
In the discrete-time framework, it is well-known that the Snell envelope is easily computed by the
backward induction
vn(i) (tn , Xtn ) = φ(i) (tn , Xtn )
(59)
and
(60)
vn(i) tj−1 , Xtj−1
(n)
n
h
(n) io
= max φn(i) tj−1 , Xtj−1 ; e−r/n E vn(i) tj , Xtj Ftj−1
,
where Ftj = σ (Xtk , k ≤ j) is the discrete-time filtration. Since the process X is Markov, the latter
conditional expectation reduces to the computation of a regression function :
n
o
n
o
(61)
E vn(i) tj , Xtj Ftnj−1 = E vn(i) tj , Xtj Xtj−1
=: ρn(i) tj−1 , Xtj−1 .
MULTIPLE AMERICAN EXERCISES
23
6.2. Computation of the Conditional Expectations. As evidenced by the above formulae, the practical implementation of this backward procedure requires the computation of many conditional expectations, and the numerical implementations will vary according to the choice made for the evaluations
of these regression functions. We briefly review the most obvious of these choices before concentrating on the method which we choose to develop.
Nonparametric Regression. At this stage of the analysis, many nonparametric regression procedures
can be brought to bear, but we refrain from attempting to reviewing them all, and we restrict ourselves
to a selected few.
(i) The Kernel Method. This well-known technique from non-parametric statistics, see e.g. Bosq [3],
has been suggested in the context of the American put option by Carrière [8]. This method is based
on the observation that for random variables or vectors A and B:
E {Aδb (B)}
(62)
E{A|B = b} =
,
E {δb (B)}
where δb is the Dirac mass at the point b. Then, given an approximate identity, i.e. a family of
functions κh which converges to δ0 when h → 0 (in some sense which we will not make precise
here), it is natural to introduce the approximation
(63)
E {Aκh (B − b)}
,
E {κh (B − b)}
and for a sample (A(s) , B (s) )1≤s≤N of N independent random vectors with the same distribution as
(A, B), the kernel estimator of the regression is given by:
1 PN
A(s) κhN (B (s) − b)
N
(64)
,
Ê{A|B = b} = 1 Ps=1
N
(s)
(s) − b)
s=1 A κhN (B
N
where hN is a sequence of positive numbers converging to zero. Despite the freedom to choose the
rate of convergence of hn to 0, the bias introduced by the approximation
of the Dirac mass by the
√
kernel function is responsible for the fact that the classical N rate of convergence of the central
limit theorem is lost. We refer to [3] for a detailed analysis of the rate of convergence of the kernel
estimator. There the interested reader will find extensions to dependent samples and proofs that this
rate decreases dramatically when the dimension of the random variable B increases.
(ii) The Basis Expansion Method. Let us assume momentarily that B is a p-dimensional random
vector, and let us denote by µB its distribution in Rp . For any square integrable random variable A,
the regression function Rp 3 b ,→ E{A|B = b} can be viewed (and characterized) as an element
of L2 (Rp , µB ). As such, it can be approximated by the partial sums of its decomposition on any
orthonormal basis of this Hilbert space. Since the coefficients in such an expansion are expectations
of products of A by functions of B, they can be estimated from a random sample (A(s) , B (s) )1≤s≤N .
This estimation technique is also standard in nonparametric regression. Its use in the context of
American option pricing was suggested by Longstaff and Schwartz [20], and the corresponding price
estimate has been shown
√ to be consistent by Clément, Lamberton and Protter [9]. As in the case of
the kernel method, the N −rate of convergence is lost because of the bias introduced by the finite
dimensional approximation. The choice of the orthonormal basis can drastically influence the rate of
24
RENÉ CARMONA AND NIZAR TOUZI
convergence. For example, it was shown by Egloff and Min-oo [11] that the rate of convergence of
this algorithm could be exponentially slow. See for example their Theorem 6.15.
Malliavin Calculus Based Simulation Method. The technique which we now consider has been
proposed by Fournié, Lasry, Lebuchoux and Lions [13], and further developed by Bouchard, Ekeland
and Touzi [4]. The asymptotic properties of the resulting numerical algorithm for the computation
of the price of American put options (and more generally, for the expected value of functions of the
solutions of reflected backward stochastic differential equations) have been analyzed by Bouchard
and Touzi [5]. The main idea is to use the Malliavin integration by parts formula in order to get rid of
the Dirac point masses in (62). In doing so one gets:
(65)
E{A|B = b} =
E{AHb (B)S}
E{Hb (B)S}
Q
where Hb (x) = pi=1 1[bi ,∞) (xi ), and S is some non-negative random variable. An important consequence of this formula is the fact that the associated Monte Carlo estimator:
1 PN
(s)
(s)
(s)
s=1 A Hb (B )S
N
(66)
Ê[A|B = b] =
,
1 PN
(s) )S (s)
H
(B
b
s=1
N
constructed from an independent sample {(A(s) , B (s) , S (s) )}s=1,··· ,N of size N , converges at the
√
N −rate by the classical central limit theorem. The following subsection is devoted to a selfcontained derivation of these facts. We use a pedestrian approach based on the log-normality of
our Gaussian framework, without ever appealing to results of the Malliavin calculus.
6.3. Integration-by-Parts based Regression Estimation. We first concentrate on a regression function of the form:
rh (x) := E{g(Wt+h )|Wt = x} =
E{g(Wt+h )δx (Wt )}
.
E{δx (Wt )}
Integration by Parts. Let us denote by ϕ the density of the standard one dimensional normal distribution, and let us assume that g is a smooth function with a bounded derivative. By the independence
of the increments of the Brownian motion, we have:
ZZ
w1
w2
dw1 dw2 .
E{g(Wt+h )δx (Wt )} =
g(w1 + w2 )δx (w1 )ϕ √ ϕ √
t
h
Integrating by parts with respect to the w1 variable, we get :
ZZ
w1
w1
w2
E{g(Wt+h )δx (Wt )} =
g(w1 + w2 )1[x,∞) (w1 ) ϕ √ ϕ √
dw1 dw2
t
t
h
ZZ
w1
w2
0
−
g (w1 + w2 )1[x,∞) (w1 )ϕ √ ϕ √
dw1 dw2 .
t
h
MULTIPLE AMERICAN EXERCISES
25
Next, we compute the second integral by integrating by parts with respect to the w2 variable. We get:
w1
w1
w2
E{g(Wt+h )δx (Wt )} =
g(w1 + w2 )1[x,∞) (w1 ) ϕ √ ϕ √
dw1 dw2
t
t
h
ZZ
w2
w2
w1
−
g(w1 + w2 )1[x,∞) (w1 ) ϕ √ ϕ √
dw1 dw2
h
t
h
(67)
= E{g(Wt+h )1[x,∞) (Wt )Sh } ,
ZZ
where the random variable
Sh =
(68)
Wt Wt+h − Wt
−
t
h
is independent of the function g. Notice that formula (67) is established for a function g ∈ Cb1 .
However, since it does not involve the regularity of g, we can conclude by a classical density argument
that it is valid whenever g(Wt+h ) ∈ L2 .
Actual Simulation. Let (W (s) )1≤s≤S be n independent samples of the Wiener process W . Then, the
Monte Carlo estimator suggested by the above formula is defined by
q̂N [g](x)
r̂N (x) :=
q̂N [1](x)
N
1 X
(s)
(s)
(s)
g(Wt+h )1[x,∞) (Wt )Sh ,
where q̂N [g](x) :=
N
s=1
(s)
where Sh is computed from the sample W (s) using formula (68). Its asymptotic properties are directly deduced from the law of large numbers and the central limit theorem for independent
identically
√
distributed random variables. In particular,√the rate of convergence is of the order N .
The price to pay in order to recover the N rate of convergence is that the variance of the estimator
q̂N [g](x) explodes as h shrinks to zero since
lim sup Sh = ∞
h&0
and
lim inf Sh = −∞.
h&0
Since our objective is to send the time step h to zero, it is necessary to find a remedy to this variance
explosion problem.
26
RENÉ CARMONA AND NIZAR TOUZI
Localization. In order to do so, we introduce a localization function. Let χ be an arbitrary smooth
function with χ(0) = 1. Following the computations leading to formula (67) we get:
E{g(Wt+h )δx (Wt )} = E{g(Wt+h )δx (Wt )χ(Wt − x)}
ZZ
w1
w2
=
g(w1 + w2 )δx (w1 )χ(w1 − x)ϕ √ ϕ √
dw1 dw2
t
h
ZZ
w1
w1
w2
=
g(w1 + w2 )1[x,∞) (w1 )χ(w1 − x) ϕ √ ϕ √
dw1 dw2
t
t
h
ZZ
w1
w2
−
g 0 (w1 + w2 )1[x,∞) (w1 )χ(w1 − x)ϕ √ ϕ √
dw1 dw2
t
h
ZZ
w1
w2
0
g(w1 + w2 )1[x,∞) (w1 )χ (w1 − x)ϕ √ ϕ √
−
dw1 dw2
t
h
ZZ
w1
w1
w2
√
√
=
g(w1 + w2 )1[x,∞) (w1 )χ(w1 − x) ϕ
ϕ
dw1 dw2
t
t
h
ZZ
w1
W2
w2
ϕ √ ϕ √
dw1 dw2
−
g(w1 + w2 )1[x,∞) (w1 )χ(w1 − x)
h
t
h
ZZ
w1
w2
0
−
g(w1 + w2 )1[x,∞) (w1 )χ (w1 − x)ϕ √ ϕ √
dw1 dw2
t
h
= E{g (Wt+h ) 1[x,∞) (Wt )Sh,χ } ,
where the random variable Sh,χ is defined by:
(69)
Sh,χ
Wt Wt+h − Wt
−
= χ(Wt − x)
t
h
0
= χ(Wt − x)Sh − χ (Wt − x)
− χ0 (Wt − x)
is again independent of the function g. For each localization function χ, one can now define a new
Monte
Carlo estimator as before. All these estimators share the nice convergence property at the
√
N -rate. Therefore, the natural question is whether one can reduce the variance of the Monte Carlo
estimator by some convenient choice of localization function χ.
4. Variance Reduction by Localization. Set G := g(Wt+h )2 , and let us consider the integrated mean
square error
Z
2
J(χ) :=
E G1Wt >x Sh,χ
dx .
R
We are interested in the integrated mean square error minimization :
(70)
V
:= min {J(χ) : χ smooth, bounded and χ(0) = 1} .
MULTIPLE AMERICAN EXERCISES
27
Using Fubini’s theorem and a path by path substitution, we get:
Z Wt
2
0
χ(Wt − x)Sh − χ (Wt − x) dx
J(χ) = E G
−∞
Z +∞
2
0
χ(y)Sh − χ (y) dy .
= E G
0
Observing that E{GSh } = 0, this provides
Z +∞
J(χ) =
E{GSh2 }|χ(y)|2 + E{G}|χ0 (y)|2 dy .
0
Hence the integrated mean square error minimization is reduced to a classical problem of calculus of
variations, which can be solved explicitly. The optimal localization function is then given by
1/2
E{GSh2 }
−ηh x
χh (x) := e
where ηh :=
(71)
.
E{G}
In particular, this shows that
ηh = O h−1/2 .
6.4. Monte Carlo Estimation for the Finite Maturity Problem. We now return to the problem
of the optimal multiple stopping problem, and more precisely to the pricing of swing options in the
framework of the discrete time approximation set up in Subsection 6.1. Let Nn be some integer
depending on the time step parameter n, and let {W (s) , 1 ≤ s ≤ Nn } be Nn independent samples of
the Wiener process. For each integer s, we denote by X (s) the process X associated to the Brownian
motion W (s) via formula (16). Also, we set:
(s)
(s)
(s)
Rh (tj , x) := Sh,χh (tj )1[x,∞) (Wtj )
i
h
(s)
(s)
(s)
(s)
(s)
= χh (Wt − x) ηh + h−1 2Wtj − Wtj −h − Wtj +h 1[x,∞) (Wtj ) ,
where Sh and χh are defined respectively in (69) and (71). Following the discussion of the previous
paragraph, we define the estimators :
(i)
(s)
(s)
1 PS
v̂
t
,
X
R
t
,
X
n
j+1
j
t
j
t
s=1
N
1/n
j+1
n
ρ̃(i)
:=
,
n tj , Xtj
(s)
1 PS
R
t
,
W
j
t
j
s=1 1/n
Nn
(i)
φ̃n tj , Xtj := φ Xtj
(s)
(s)
(i−1)
1 PS
t
,
X
v̂
t
+
δ,
X
R
n
j
j
t
j
s=1
t
+δ
δ
N
j
n
1tj ≤1−δ
+e−rδ
(s)
1 PS
R
t
,
W
j
t
j
s=1 δ
Nn
(i)
(i)
of ρn tj , Xtj and φn tj , Xtj respectively. These
estimatorsare defined inductively, given the es(i)
(s)
(i−1)
timators v̂
(., .) and the previous estimator v̂n tj+1 , Xtj+1 in the backward procedure. Finally,
28
RENÉ CARMONA AND NIZAR TOUZI
(i)
(i)
we observe that ρn ≤ iK and φn ≤ iK. Hence, in order to ovoid an explosion of the algorithm, we
define the truncated estimators (see [5]) :
+
ρ̂(i)
:= (iK) ∧ ρ̃n(i) tj , Xtj
,
n tj , Xtj
+
φ̂(i)
:= (iK) ∧ φ̃(i)
,
n tj , Xtj
n tj , Xtj
and
(72)
v̂n(i) tj , Xtj
n
o
:= max ρ̂n(i) tj , Xtj , φ̂(i)
t
,
X
.
j
t
n
j
According to the error estimate of [5], in order for the approximation error to be of the order of n−1/2 ,
one has to choose a number Nn of simulated trajectories such that
Nn = O n7/2 .
The Value Functions. The above algorithm was implemented and tested for the swing put option
with the following characteristics: maturity T = 1 year, refraction period δ = 0.1, r = .05, σ = .30,
maximal number of exercise rights ` = 5, n = 50.
F IGURE 6. Graphs of the functions v (1) (t, · ) for t = .59, .58, · · · , .02, .01 (left) and
of the functions v (3) (t, · ) for t = .49, .47, · · · , .02, .01 (right).
The left pane of Figure 6 gives the plots of the graphs of the functions x ,→ v (1) (t, x) for t =
.59, .58, · · · , .02, .01. Two remarks are in order. First, these graphs are not computed over the same
range of values of x. Essentially, we computed the values of v (k) (t, x) for the values of x which can
be reached by the sample paths of the diffusion process Xt , and we determined this range of values of
x from the results of our simulations. The second remark concerns the noise in the numerical results.
Obviously, we should expect zero in the right hand side of the plots, and we see quite significant
departures from this expectation. The right pane of the figure gives the plots of the graphs of the
functions x ,→ v (3) (t, x) for t = .59, .58, · · · , .02, .01. for t = .49, .47, · · · , .02, .01.
MULTIPLE AMERICAN EXERCISES
29
F IGURE 7. Surface plot of the graph of the function v (1) when regarded as function
of both t and x. The variable TAU represents 100 ∗ (T − t).
Figure 7 gives the same plot as the left pane of Figure 6, but instead of super-imposing the onedimensional graphs on the same plot, we use both the t and the x variables to produce surface plots,
or to be more specific the scaled time to maturity τ = 100(T − t) and x. The fact that the range
of x varies with t is obvious from this surface plot, and as expected, it is limited by some form of
parabola. Plotting the graphs of the other value functions v (k) would produce very similar results and
we refrain from producing them.
Number of Monte Carlo Scenarios. We present some partial numerical results to illustrate the effect
of the number of trajectories Nn . According to the result of [5] which we re-derived above, the
number Nn should be of the order of n7/2 . The results collected in the following Table 1 show that a
very high precision can be achieved even with a significantly smaller number of simulated trajectories.
Table 1. Swing put option values for various numbers of simulations
T = 1 year, δ = 0.1 year, S0 = K = 100, r = .05, σ = .30, n = 50
30
RENÉ CARMONA AND NIZAR TOUZI
v (1) [stand.
v (2) [stand.
v (3) [stand.
v (4) [stand.
v (5) [stand.
dev.]
dev.]
dev.]
dev.]
dev.]
N = 8, 192
9.84 [.22%]
19.21 [.56%]
28.69 [.68%]
38.34 [.57%]
48.17 [.50%]
N = 16, 384
9.85 [.12%]
19.26 [.30%]
28.80 [.30%]
38.48 [.27%]
48.32 [.30%]
Exercise Regions. Next we identify an estimate of the exercise region for each of the value functions v (i) considered as a single stopping problem associated to the reward function φ(i) (t, x). The
corresponding exercise boundaries are given by the graphs of the functions t ,→ x̂∗i (t). Estimates
of these boundaries computed with the Monte Carlo method described in this section are plotted in
Figure 8. The computations were performed with the following parameters: maturity T = 10 months,
refraction period δ = 2 months, r = .05, σ = .30, maximal number of exercise rights ` = 5, n = 50,
N = 8192. As expected these exercise boundaries are increasing functions of the time-to-maturity
variable. We also verify that x̂∗i (t) ≥ x̂∗i−1 (t). This property is consistent with the intuition. We
proved rigorously this result in the case of the perpetual put options in Lemma 5, but a proof of this
fact in the finite maturity case is still lacking: this monotonicity remains an interesting open problem.
F IGURE 8. Estimates of the boundaries of the exercise regions of swing options with
` = 5 exercise rights, as given by the graphs of the functions t ,→ x∗i (t) computed
via the Monte Carlo procedure described in the text.
MULTIPLE AMERICAN EXERCISES
31
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